Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 172 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 625 (d) 425
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
5. 5.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 25 (c) 5 (d) 73
6. 6.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 6 (d) 1 and 12
7. 7.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 1 and 5 (d) 2 and 5
8. 8.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4875 (d) 4225
9. 9.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 11 (d) 35
10. 10.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
11. 11.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3136 (c) 4106 (d) 3616
12. 12.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 36 (c) 13 (d) 37
13. 13.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
15. 15.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
16. 16.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
18. 18.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
19. 19.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
20. 20.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
21. 21.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
22. 22.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
23. 23.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 9801 (d) 10201
24. 24.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 6 and 9 (d) 1 and 14
25. 25.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9801 (d) 9811
26. 26.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4624 (c) 5194 (d) 4904
27. 27.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
28. 28.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 63 (d) 1 and 15
29. 29.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 2025 (d) 3025
30. 30.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
31. 31.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 7225 (d) 8125
32. 32.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
33. 33.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7235 (d) 7225
34. 34.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
35. 35.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 10 (d) 7
36. 36.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2601 (c) 2401 (d) 2499
38. 38.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
39. 39.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 25 (c) 81 (d) 53
40. 40.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9

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